Abstract truncations - An Introduction To Homological Algebra

In this section we introduce abstract truncation functors for a t-structure. These will be needed in proving that the core of a t-structure is an abelian category. In this sectionDdenotes a triangulated category and t“ pD^ď0,D^ě0q a fixed t-structure onD.

Let us construct the abstract truncation functors. For any X PObDfix a distinguished triangle

A X B Ar1s

whereAPD^ď0and BPD^ě1, given by T3, and letτď0X “A andτě1X “B. For any integernPZ, we define τďnX “ pτď0pXrnsqqr´ns PD^ďn and τěn`1“ pτě1pXrnsqqr´ns PD^ěn`1.

Using these definitions, and by application of rotation TR3, one obtains that for any n P Z the following is a distinguished triangle

τďnX X τěn`1X pτďnXqr1s (7.1)

Indeed, by definition ofτď0andτě1 the following is a distinguished triangle

τď0pXrnsq Xrns τě1pXrnsq pτď0pXrnsqqr1s By rotation, TR3, we get that

τď0pXrnsqr´ns Xrnsr´ns τě1pXrnsqr´ns pτď0pXrnsqqr´n`1s

is a distinguished triangle. Since translation is an additive automorphism, Xrnsr´ns “X, and we have obtained that (7.1) is a distinguished triangle.

Letf :X ÑY be any morphism inD. Then by lemma 7.1.3 and lemma 7.1.4, the following diagram is a unique morphism of distinguished triangles

τďnX X τěn`1X pτďnXqr1s

τďnY Y τěn`1Y pτďnXqr1s

τďnf f τ_ěn`1f fr1s

for anynPZ. We define the morphismτďnf andτěn`1f as in the diagram. By using uniqueness, one easily verifies thatτďn:DÑD^ďn andτěn`1:DÑD^ěn`1 are functors.

The following proposition shows that the distinguished triangles, given by T3, are isomorphic up to unique isomorphism. In particular, the functorsτďn andτěn`1are unique up to unique isomorphism.

Proposition 7.2.1. Suppose

A X B Ar1s (7.2)

is a distinguished triangle, withAPObD^ďn andBPObD^ěn`1, then it is uniquely isomorphic to the distinguished triangle (7.1).

Proof. Since Mor_DpτďnX, Bq “0 and Mor_DpA, τěn`1Xq “0 by lemma 7.1.3, lemma 7.1.4 implies that there exists a unique morphismsφandψbetween (7.1) and (7.2), because Mor_DpτďnX, Br´1sq “0 and Mor_DpA,pτěn`1Xqr´1sq

“0 by lemma 7.1.3. By lemma 7.1.4 and lemma 7.1.3 the only automorphisms of distinguished triangles (7.1) and (7.2) are the identity morphisms. This shows that ψφ and φψ are identity morphisms, hence ψ “φ^´1, and the distinguished triangles are canonically isomorphic.

Lemma 7.1.4 and lemma 7.1.3 imply that for any m ď n there are unique morphisms τďmX Ñ τďnX and τěmX ÑτěnX, because of the following unique morphism of distinguished triangles

τďmX X τěm`1X pτďmXqr1s

τďnX X τěn`1X pτďnXqr1s

(7.3)

Lemma 7.2.2. The following conditions are equivalent for any objectX ofDand for any integernPZ. (i) X PD^ďn.

(ii) The morphismτďnX ÑX is an isomorphism.

(iii) τěn`1X “0.

Proof. piq ñ piiq: By lemma 7.1.3, the morphismXÑτěn`1X is the zero morphism. Thus by TR3 and TR5 we have the following unique, by lemma 7.1.4, morphism of distinguished triangles

X X 0 Xr1s

τďnX X τěn`1X pτďn´1Xqr1s

ψ ψr1s

Also, we have the following unique, by lemma 7.1.4, morphism of distinguished triangles τďnX X τěn`1X pτďn´1Xqr1s

X X 0 Xr1s

ψ¹ φr1s

whereψ¹ is the morphismτďnX ÑX by commutativity. Composite of these morphisms, in both order, is a unique morphism of distinguished triangles, by lemma 7.1.4, so we haveψ¹ψ“IdX andψψ¹ “IdτďnX. HenceX –τďnX. piiq ñ piiiq: This follows from 4.1.6 applied to the following morphism of distinguished triangles obtained by TR5

τďnX τďnX 0 pτďnXqr1s

τďnX X τěn`1X pτďnXqr1s

piiiq ñ piq: Apply 4.1.6 to the following morphism of distinguished triangles obtained by TR3 and TR5

X X 0 pτď0Xqr1s

τďnX X 0 pτďnXqr1s

We have also the dual version of the previous lemma.

Proposition 7.2.3. For any objectX PDand nPZthe following conditions are equivalent.

(i) X PD^ěn`1.

(ii) The morphismXÑτěn`1X is an isomorphism.

(iii) τďnX “0.

Proof. piq ñ piiq: The morphismτďnX ÑX is 0 by lemma 7.1.3. Hence by TR5 we have the following morphisms of distinguished triangles

τďnX X τěn`1X pτďnXqr1s

0 X X 0

τďnX X τěn`1X pτďnXqr1s

ψ¹

where ψ¹ is the morphism X Ñτěn`1X by commutativity. The composites, in both order, of the two morphisms are unique by lemma 7.1.4, so IdX “ψψ¹ and Idτ_ěn`1X “ψ¹ψ. This shows thatX –τěn`1X.

piiq ñ piiiq: This follows from corollary 4.1.6 applied to the following morphism of distinguished triangles obtained by TR3 and TR5

0 X X 0

τďnX X τěn`1X pτďnXqr1s

piiiq ñ piq: Apply 4.1.6 to the following morphism of distinguished triangles obtained by TR5

0 X X pτď0Xqr1s

0 X τěn`1X 0

We are ready to prove that τďn and τěn`1 are adjoints to the corresponding inclusion functorsiďn andiěn`1, respectively.

Proposition 7.2.4(Adjoints). The functorsτďn andτěn`1are the right and left adjoints to the inclusion functors iďn :D^ďnÑD andiěn`1:D^ěn`1ÑD for allnPZ.

Proof. To prove the theorem, we will need the following isomorphisms

Φⁿ_X,Y : Mor_DpX, YqÑ^– Mor_D^ďnpX, τďnYq Ψ^n`1_Z,W : Mor_D^ěn`1pτěn`1Z, WqÑ^– Mor_DpZ, Wq

for anyX PObD^ďn, W PObD^ěn`1, andY, Z PObD. To construct these maps, let h:X ÑY be a morphism in Dand letk¹:τěn`1Z ÑW be a morphism inD^ěn`1. By TR1 to TR3 and TR5 and lemma 7.1.4 we have the following unique morphisms of distinguished triangles inD

X X 0 Xr1s

τďnY Y τěn`1Y pτďnYqr1s

h¹ h hr1s

φY

τďnZ Z τěn`1Z pτďnZqr1s

0 W W 0

ψ_Z

k k¹

We define Φⁿ_X,Yphq “h¹ and Ψ^n`1_Z,Wpk¹q “k. By proposition 4.1.5, applied to the distinguished triangle (7.1) forY andZ, we have the following exact sequences

Mor_DpX,pτěn`1Yqr´1sq Mor_DpX, τďnYq ^pφ^Y^q^˚ Mor_DpX, Yq Mor_DpX, τěn`1Yq and

Mor_DppτďnZqr1s, Wq Mor_Dpτěn`1Z, Wq ^pψ^Z^q Mor_DpZ, Wq Mor_DpτďnZ, Wq

where Mor_DpX,pτěn`1Yqr´1sq “ Mor_DpX, τěn`1Yq “ 0 and Mor_DpτďnZ, Wq “ Mor_DppτďnZqr1s, Wq “ 0, by lemma 7.1.3. Hence from exactness it follows that Φⁿ_X,Y and Ψ^n`1_Z,W are isomorphisms.

τďn right adjoint to iďn: To show thatτďn is right adjoint toiďn, by theorem 1.3.4, it suffices to show that for any morphismsf :X¹ ÑX ofD^ďn andg:Y ÑY¹ ofDthe following diagram is commutative

Mor_DpX, Yq Mor_D^ďnpX, τďnYq

Mor_DpX¹, Y¹q Mor_D^ďnpX¹, τďnY¹q

Φⁿ_X,Y

MorDpf,gq MorDpf,τ_ďngq Φⁿ_X1,Y1

(7.4)

LethPMor_DpX, Yq. Then Mor_Dpf, gqphq “ghf. Now Φⁿ_X1,Y¹pghfqis the unique morphism k:X¹ÑτďnY¹ such thatφ¹k“ghf, by lemma 7.1.4. This means that we have the following unique morphism of distinguished triangles

X¹ X¹ 0 X¹r1s

τďnY¹ Y¹ τěn`1Y¹ pτěn`1Y¹qr1s

k ghf

φ¹

The map ΦX,Y sendsh to the unique morphisml : X Ñ τďnY such thatφl “h. Now Mor_Dpf, τďngqplqis the compositepτďngqlf. Hence we have the following unique morphism of distinguished triangles

X¹ X¹ 0 X¹r1s

X X 0 Xr1s

τďnY Y τěn`1Y pτěn`1Yqr1s

τďnY¹ Y¹ τěn`1Y¹ pτěn`1Y¹qr1s

f f

l h

τďng g pτďngqr1s

φ¹

By lemma 7.1.4, the morphism of distinguished trianglespX¹, X¹,0q Ñ pτďnY¹, Y¹, τěn`1Y¹q, is unique. There-forek“τď0pgqlf. This shows that the diagram (7.4) is commutative.

τěn`1 is left adjoint toiěn`1: By theorem 1.3.4 we have to show that for all morphismsf :X ÑX¹ of D^ěn`1 andg:Y¹ÑY ofD, the following diagram is commutative.

Mor_D^ěn`1pτěn`1pYq, Xq Mor_DpY, Xq

Mor_D^ěn`1pτěn`1pY¹q, X¹q Mor_DpY¹, X¹q

ΨY,X

Morpτěn`1pgq,fq Morpg,fq Ψ_Y1,X1

(7.5)

Let hPMor_D^ěn`1pτěn`1pYq, Xq. Then ΨY¹,X¹pf hτěn`1pgqqis the unique morphismk:Y¹ ÑX¹ such that the following diagram is a unique morphism of distinguished triangles

τďnpY¹q Y¹ τěn`1pY¹q pτďnpY¹qqr1s

0 X¹ X¹ 0

k f hτ_ěn`1pgq

The morphism ΨY,Xphq is the unique morphism l:Y ÑX such that the following is a morphism of distin-guished triangles

τďnY Y τěn`1Y pτďn`1Yqr1s

0 X X 0

l h

Now Mor_Dpf, gqplq is the unique morphism such that the following diagram is a morphism of distinguished triangles

τďnpY¹q Y¹ τěn`1pY¹q pτďnpY¹qq

0 X¹ X¹ 0

f lg f hτ_ěn`1pgq

By uniqueness of proposition 7.2.1, k“f lg. This shows that the diagram (7.5) is commutative.

We have the following isomorphisms for abstract truncations.

Proposition 7.2.5. For any integersmďnwe have canonical isomorphisms of functors τďmτďn –τďm–τďnτďm τěmτěn–τěn –τěnτěm.

Proof. τďmτďn–τďm: By proposition 7.2.4 and lemma 1.3.2 it suffices to show that τďmτďn is right adjoint to the inclusion functor iďm:D^ďmÑD. Recall from the proof of proposition 7.2.4 that we have the following isomorphism

ΦⁿX,Y : Mor_DpX, YqÑ^„ Mor_D^ďnpX, τďnYq,

for any nP Z. Let Φ¹_X,Y “ Φ^m_X,τ_ďn_Y ˝Φⁿ_X,Y. By theorem 1.3.4, it suffices to show that for all morphisms f :X¹ ÑX ofD^ďm andg:Y ÑY¹ ofDthe following diagram commutes

Mor_DpX, Yq Mor_D^ďmpX, τďmτďnYq

Mor_DpX¹, Y¹q Mor_D^ďmpX¹, τďmτďnY¹q

Φ¹_X,Y

MorDpf,gq MorDďmpf,τďm τďn gq

Φ¹_X1,Y1

But by the proof of proposition 7.2.4 the left and right squares of the following diagram are commutative there exists the following unique morphism of distinguished triangles

τďmτďnX τďmτďnX 0 pτďmτďnXqr1s

τďmX X τěm`1X pτďmXqr1s

where τďmτďnXÑτďmX is the isomorphism.

τěnτěm»τěn: The argument is similar as in the previous case. Let Ψ¹_X,Y “Ψⁿ_τ_ěm_X,Y ˝Ψ^m_X,Y. These functors are defined in the proof of proposition 7.2.4. By lemma 1.3.2 we need to show that τěnτěm is the left adjoint of iěn. By theorem 1.3.4, it suffices to show that for all morphismsf :X¹ ÑX ofDand g :Y ÑY¹ of D^ěn

But by the proof of proposition 7.2.4 both of the squares in the following diagram are commutative Mor_D^ěnpτěnτěmX, Yq Mor_D^ěmpτěmX, Yq Mor_DpX, Yq X toiěn. Hence we have the following unique morphism of distinguished triangles whereτěnXÑτěnτěmX is the isomorphism

τďn´1X X τěnX pτěn´1Xqr1s

0 τěnτěmX τěnτěmX 0

τďnτďm–τďm: The morphism τďnτďmX ÑτďmX is an isomorphism by lemma 7.2.2, because τďmX PD^ďmĂ D^ďn, by lemma 7.1.2. Let f : X Ñ Y be a morphism in D. Then commutativity of the left square of the following morphism of distinguished triangles shows that the functors are isomorphic.

τďnτďmX τďmX τěn`1τďmX pτďnτďmXqr1s

τďnτďmY τďmY τěn`1τďmY pτďnτďmYqr1s

–

τ_ďnτ_ďmf τ_ďmf –

τěmτěn–τěn: By proposition 7.2.3, the morphismτěmτěnX ÑτěnX is an isomorphism, becauseτěnX PD^ěnĂ D^ěmby lemma 7.1.2. Letf :X ÑY be a morphism in D. Then commutativity of the middle square of the following diagram shows thatτěmτěn»τěn.

τďm´1τěnX τěnX τěmτěnX pτďm´1τěnXqr1s

τďm´1τěnY τěnY τěmτěnY pτďm´1τěnYqr1s

–

τ_ěnf τ_ěmτ_ěnf –

Lemma 7.2.6. Let

A X B Ar1s

be a distinguished triangle withA, BPD^ďn. ThenX PD^ďn. Similarly, if A, BPD^ěn`1, thenX PD^ěn`1. Proof. LetA, BPD^ďn. By lemma 7.2.2 it suffices to show thatτěn`1X “0. Consider the following exact sequence

Mor_DpB, τěn`1Xq Mor_DpX, τěn`1Xq Mor_DpA, τěn`1Xq given by proposition 4.1.5 applied to the given distinguished triangle. Then

Mor_DpB, τěn`1Xq “Mor_DpA, τěn`1Xq “0

by lemma 7.1.3. Recall that Ψ^n`1_X,X : Mor_D^ěn`1pτěn`1X, τěn`1Xq Ñ Mor_DpX, τěn`1Xq, defined in the proof of proposition 7.2.4, is an isomorphism. By exactness Mor_DpX, τěn`1Xq “0. Hence

Mor_D^ěn`1pτěn`1X, τěn`1Xq “0 Now Idτěn`1X “0, and soτěn`1X “0.

Suppose that A, B PD^ěn`1. By proposition 7.2.3 it suffices to show thatτďnX “0. By proposition 4.1.5 we have the following exact sequence

Mor_DpτďnX, Aq Mor_DpτďnX, Xq Mor_DpτďnX, Bq

where Mor_DpτďnX, Aq “Mor_DpτďnX, Bq “0 by lemma 7.1.3. Now Φⁿ_X,τ_ďn_X: Mor_DpτďnX, Xq ÑMor_DpτďnX, τďnXqis an isomorphism. Thus by exactness and Φⁿ_X,τ_ďn_X, Idτ_ďnX“0. This completes the proof.

Proposition 7.2.7. For anyn, mPZwe have

τěmτďn–τďnτěm.

Proof. Supposemąnand letX PObD. Then by lemma 7.1.2τďnXPD^ďm´1, soτěmτďnX “0 by lemma 7.2.2.

Similarly, by lemma 7.1.2,τěmX PD^ěn`1. HenceτďnτěmX “0 by proposition 7.2.3. Therefore we may assume thatmďn.

LetX PObDand consider the distinguished triangle (7.1) forτěmX

τďnτěmX τěmX τěn`1τěmX pτďnτěmXqr1s

By TR3 and lemma 7.2.6,τďnτěmX PD^ěm, becausepτěnτěmXqr´1s – pτěmτěn`1Xqr´1s PD^ěm`1ĂD^ěm, by lemma 7.1.2 and proposition 7.2.5, andτěn`1τěmX –τěmτěn`1X PD^ěm, by proposition 7.2.5.

Recall the definition of the maps Φⁿ_X,Y and Ψⁿ_X,Y in the proof of proposition 7.2.4. We have the following composition of isomorphisms φ:τěmτďnX ÑτďnτěmX obtained by first taking by lemma 7.1.4 the following unique morphism of distinguished triangles

τďnX τďnX 0 pτďnXqr1s

τďnτěmX τěmX τěn`1τěmX pτďnτěmXqr1s

l (7.7)

and then use the morphismlto get the following unique, by lemma 7.1.4, morphism of distinguished triangles τďm´1τďnX τďnX τěmτďnX pτďm´1τďnXqr1s

0 τďnτěmX τďnτěmX 0

l φ (7.8)

To show thatφis an isomorphism, consider the following upper cap diagram

τěn`1X X

The lower triangle in the diagram is distinguished because by proposition 7.2.5 we have the following isomorphism of triangles

τďm´1τďnX τďnX τěmτďnX pτďm´1τďnXqr1s

τďm´1X τďnX τěmτďnX pτďm´1Xqr1s

– –

In particular, the morphism τďm´1X ÑτďnX in the diagram is the unique morphism given in (7.3) by the proof of proposition 7.2.5. Hence the right triangle in the upper cap is commutative.

Complete the upper cap to the following lower cap

τěn`1X X

X¹

τěmτďnX τďm´1X

r1s

ö d

By proposition 7.2.1,X¹ –τěmX. By commutativity of the upper triangle,τěmX Ñτěn`1X is the unique mor-phism given by (7.3). We have the following unique mormor-phism of distinguished triangles, obtained by lemma 7.1.4, where the morphismτěn`1X Ñτěn`1τěmX is an isomorphism by proposition 7.2.5

τďnX X τěn`1X pτďnXqr1s

0 τěn`1τěmX τěn`1τěmX 0

–

Here both of the morphisms X Ñ τěn`1X and X Ñ τěn`1τěmX factor through the morphism X Ñ τěmX by (7.3). From this we get the following isomorphism of triangles

τěmτďnX τěmX τěn`1X pτěmτďnXqr1s

τěmτďnX τěmX τěn`1τěmX pτěmτďnXqr1s

–

whereτěn`1τěmX Ñ pτěmτďnXqr1sis the composite

τěn`1τěmXÑτěn`1XÑ pτěmτďnXqr1s. (7.9) In particular, pτěmτďnX, τěmX, τěn`1τěmXq is a distinguished tringle. Therefore by TR3 and TR5 and corol-lary 4.1.6, we have a unique isomorphismδpXqwhich makes the following diagram an isomorphism of distinguished triangles

τěmτďnX τěmX τěn`1τěmX pτěmτďnXqr1s

τďnτěmX τěmX τěn`1τěmX pτďnτěmXqr1s

δpXq δpXqr1s (7.10)

To verify that φ “ δpXq, it suffices, by lemma 7.1.4, to show that l “ δpXq ˝k. By commutativity of the octahedra and (7.10) we have the following equalities

τďnXÑ^k τěmτďnX ^δpXqÑ τďnτěmX ÑτěmX“τďnX Ñ^k τěmτďnXÑτěmX

“τďnX ÑX ÑτěmX.

Since Ψⁿ_τ_ďn_X,τ_ěm_X is an isomorphism, we haveδpXq ˝k“l. ThereforeδpXq “φ.

To show thatδis an isomorphism of functors, we need to verify that for any morphismf :X ÑY the following diagram is commutative

τěmτďnX τďnτěmX

τěmτďnY τďnτěmY

δpXq

τěmτďnpfq τďnτěmpfq δpYq

By lemma 7.1.4 we have the following unique morphism of distinguished triangles.

τďnX X τěn`1X pτďnXqr1s

τďnY Y τěn`1Y pτďnYqr1s

τ_ďnf f τ_ěn`1f pτ_ďnfqr1s (7.11)

The compositeτďnτěmpfq ˝δpXqis the unique, by lemma 7.1.4, morphism such that the following composite is a morphism of distinguished triangles

τěmτďnX τěmX τěn`1τěmX pτěmτďnXqr1s

τďnτěmX τěmX τěn`1τěmX pτďnτěmXqr1s

τďnτěmY τěmY τěn`1τěmY pτďnτěmYqr1s

δpXq

τďnτěmpfq τěmpfq

(7.12)

The composite δpYq ˝τěmτďnpfq is the unique morphism such that the following composite is the unique, by lemma 7.1.4, morphism of distinguished triangles

τďm´1τďnX τďnX τěmτďnX pτďm´1τďnXqr1s

τďm´1τďnY τďnY τěmτďnY pτďm´1τďnYqr1s

0 τďnτěmY τďnτěmY 0

τ_ďm´1τ_ďnpfq τ_ďnpfq τ_ěmτ_ďnpfq

l δpYq“φ

(7.13)

To show thatτďnτěmpfq ˝δpXq “δpYq ˝τěmτďnpfq, it suffices, by the fact thatpΨ^m_X,τ_ďn_τ_ěm_Yq^´1is an isomorphism, to show that

τďnX ÑτěmτďnX^δpXqÑ τďnτěmX ^τ^ďn^τÑ^ěm^pfqτďnτěmY

“τďnX ÑτěmτďnX^τ^ěm^τÑ^ďn^pfqτěmτďnY ^δpYÑ^q“φτďnτěmY.

(7.14)

By commutativity of (7.13) we have

τďnXÑτěmτďnX ^τ^ěmÑ^τ^ďn^pfqτěmτďnY ^δpYÑ^q“φτďnτěmY “τďnX ^τ^ďnÑ^pfqτďnY Ñ^l τďnτěmY.

To show equality (7.14), by the fact thatpΦ^m_τ_ďn_X,τ_ďn_τ_ěm_Yq^´1 is an isomorphism, it suffices to show that τďnX ÑτěmτďnX ^δpXqÑ τďnτěmX ^τ^ďn^τÑ^ěm^pfqτďnτěmY ÑτěmY “τďnX ^τ^ďnÑ^pfqτďnY Ñ^l τďnτěmY ÑτěmY.

By commutativity of (7.7) forY and (7.11), we have

τďnX^τ^ďnÑ^pfqτďnY Ñ^l τďnτěmY ÑτěmY “τďnX ^τ^ďnÑ^pfqτďnY ÑY ÑτěmY

“τďnX ÑX Ñ^f Y ÑτěmY.

Now

τďnX ÑτěmτďnX^δpXqÑ τďnτěmX ^τ^ďn^τÑ^ěm^pfqτďnτěmY ÑτěmY

“τďnX ÑτěmτďnX^δpXqÑ τďnτěmX ÑτěmX ^τ^ěmÑ^pfqτěmY (7.12)

“τďnX ÑτěmτďnXÑτěmX ^τ^ěmÑ^pfqτěmY (7.12)

“τďnX ÑXÑτěmX^τ^ěmÑ^pfqτěmY (7.7)

“τďnX ÑXÑ^f Y ÑτěmY. (7.11)

This shows thatτěmτěnpfq ˝δpXq “δpYq ˝τěmτďnpfqand completes the proof.

In document An Introduction To Homological Algebra (sivua 140-151)